Eddie Nketiah's record-breaking 14th Under-21s goal and Huseyin Turkmen's own goal gave the hosts the victory they needed to reach next year's tournament with two games to spare.
Aaron Ramsdale saved Halil Dervisoglu's penalty during a nervy second half for the hosts at Molineux.
England never found their rhythm and Nketiah hit a post with his own late spot-kick before his historic goal, and Boothroyd wants improvement.
"I thought after the first goal it was going to be a relatively easy night but it was anything but," he said.
"We managed to get ourselves through so we're delighted and we can use the next two games as preparation.
"But I said to the players we have to be better than that if we want to progress further. I just don't want too many of these games where we didn't do too well but still win.
"It was a more senior team and sometimes you can have brilliant weeks of training and it doesn't go according to plan.
"With the high standards we set we have to be better. We had a discussion in the dressing room about how we defend better."
Boothroyd made eight changes from the 3-3 draw in Andorra last week with a stronger side initially bossing the game.
They opened the scoring after 17 minutes when Ryan Sessegnon's cross was turned in by Turkmen.
But the Young Lions struggled to build on their lead and Ramsdale saved Ahmet Canbaz's drive just before the break.
He was called into action again eight minutes after half-time when Turkey were awarded a penalty.
Dervisoglu latched onto Oliver Skipp's poor backpass to poke past Ramsdale and, despite there being no contact, went down after losing the ball.
Referee Willy Delajod awarded the penalty but Ramsdale saved Dervisoglu's tame effort.
Cemali Sertel was then denied by Ramsdale with England second best, but they were given the chance to make it 2-0 with seven minutes left when James Justin went down in the area only for Nketiah to hit the outside of the right-hand post from 12 yards.
But he got his goal with two minutes left, beating Altay Bayindir after being sent clear by Sessegnon to pull clear of Alan Shearer and Francis Jeffers.
Dervisoglu pulled a goal back in stoppage time but England, top of Group Three, reached next year's tournament in Hungary and Slovenia.
Boothroyd added: "Eddie's a huge talent, a genuinely nice person. He's great off the pitch for us and on the pitch he takes no prisoners. He's a real warrior. To surpass Alan is a huge achievement."
Join our new commenting forum
Join thought-provoking conversations, follow other Independent readers and see their replies